Independent random variables are those variables of a random experiment whose values does not affect the value of the other variable. Two random variables $X$ and $Y$ are independent if, P$(X|Y)$ = $P(X)$, that is, the probability of $X$ if $Y$ has already occurred is not affected at all. The random variables $X$ and $Y$ are independent of each other iff,

$P(X$ and $Y)$ = $P(X)P(Y)$

Word Problems

Problem 1: 

Aren tossed two dices, one was green in color and other one was black in color. Show that their outcomes are independent of each other.

Solution: 

The sample space of throwing a dice = $\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$

The probability of getting any number in green dice = $\frac{1}{6}$

The probability of getting any number in black dice = $\frac{1}{6}$

The probability of getting any number in green and black dice = $\frac{1}{36}$

Hence, $P$(green and black) = $P$(green)$P$(black) and so these outcomes are independent random variables.
Problem 2:

A variable $X$ represents the number of defected pens in a box, and variable $Y$ represents the number of pens having more than two defective pens when there are two boxes having one pen. Given below is the probability distribution of two random variables, $X$ and $Y$. Find out if they are independent random variables.

   X=0   X=1 
 Y=1   0.1  0.2
 Y=2  0.1  0.2

Solution:

From the given table of joint probability distributions, we can find different probabilities.

The cumulative distribution can be given in the following table.

   X=0
 X=1
 
 Y=1
 0.1  0.2  0.3 
 Y=2  0.1  0.2  0.3
   0.2  0.4  

Now, $P((X$ = $0)$ and $(Y$ = $1))$ = $0.1$

$P(X$ = $0)P(Y$ = $1)$ = $0.2\ \times\ 0.3$ = $0.06$.

Hence, $P((X$ = $0)$ and $(Y$ = $1))\ \neq\ P(X$ = $0)P(Y$ = $1)$.

$X$ and $Y$ are not independent variables.
Problem 3:

The variable A defines the probability of a bulb to be on or off, and Random variable B gives the probability if room is locked or not. Are these variables independent if the joint probability distribution is given as below.

   A = Yes  A = No
 B = Yes
 0.25  0.25
 B = No  0.25  0.25

Solution:

Following table will give the probability distribution for these equally likely events.

   A = yes  A = No  
 B = Yes
 0.25  0.25  0.5
 B = No
 0.25  0.25  0.5
   0.5  0.5  

$P((B$ = $yes)and(A$ = $yes))$ = $0.25$ = $P(B$ = $yes)\ \times\ P(A$ = $yes)$

$P((B$ = $yes)and(A$ = $no))$ = $0.25$ = $P(B$ = $yes)\ \times\ P(A$ = $no)$

$P((B$ = $no)and(A$ = $yes))$ = $0.25$ = $P(B$ = $no)\ \times\ P(A$ = $yes)$

$P((B$ = $no)and(A$ = $no))$ = $0.25$ = $P(B$ = $no)\ \times\ P(A$ = $no)$

For all events, $P(B$ and $A)$ = $P(B)P(A)$ and hence, $A$ and $B$ are independent random variables.